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waves on steep (1:1) and gentle (1:19.85) slopes. The solid line in Fig. 5.13 shows
the dependence corresponding to the run-up height on a vertical wall, calculated by
formula (5.46). In the case of a run-up on a steep slope, the dependence of
R
(
ξ
0
) is
actually very close to linear. But in the case of the run-up of waves of large amplitude
on a gentle slope deviation is seen of the dependence from linearity towards a de-
crease in the run-up height. The bend in the dependence, observed at
ξ
0
H
∼
0
.
03
(Fig. 5.13b) corresponds to transition to run-ups involving wave breaking.
In Fig 5.13a. it is seen that the run-up on a steep slope is approximately twice
the wave amplitude far from the coast, which complies with the theoretical result
for the run-up on a vertical wall (5.46). In the case of a run-up on a gentle slope
(Fig. 5.13b) the height increases noticeably; here, it is three to four times higher
than the wave amplitude far from the coast. In any case, the energy losses, related
to wave breaking, result in a reduction of the run-up height. In certain conditions
the run-up height on a gentle slope, involving wave breaking, may even turn out to
be smaller than the same quantity in the case of a vertical wall.
The data of numerical simulation, presented in Fig. 5.13, are in good accordance
with the results of laboratory experiments. Details of the numerical algorithm are
described in [Titov, Synolakis (1995)].
In conclusion of this section we shall dwell upon certain difficulties arising in nu-
merical simulation of a tsunami run-up with account of the real relief of the coastal
area. The
first difficulty
is related to the absence of or insufficiently detailed bathy-
metric and topographical data. For modelling tsunamis in the open ocean, where
wavelengths are significant, of the order of 100 km, the existing global data, for
example, ETOPO-2 with a resolution of 2 angular minutes (
4 km) are quite suffi-
cient. But for reliable numerical simulation of the tsunami dynamics in the coastal
zone it is necessary to have data on the reliefs of the bottom and of the coastal
area with a space resolution hundreds of times better (
∼
10 m). This requirement is
related not only to the significant reduction of wavelengths in shallow water. The
quality of topographical data directly influences the precision in resolving the prac-
tical problem—determination of the run-up boundaries. Here, it must also be noted,
that for resolving the run-up problem accurately it is also necessary to have at one's
disposal information on tidal-level oscillations.
The most reliable criterion of applicability of one or another tsunami run-up
model consists in practical tests. Laboratory experiments, naturally, permit to judge
the efficiency of numerical models, but in any case the most reliable test consists
in comparison of the results of simulation with data on real tsunamis. Here, we
encounter the
second difficulty
, related to the existence and quality of results of
in situ measurements. To test a model detailed measurements of the run-up area are
required, desirably supplemented with information on the water flow parameters
on the coast. In recent years, the database of run-up parameters is regularly up-
graded with high-precision measurements performed by international expeditions,
for which the investigation of coastal areas hit by tsunamis is mandatory. Contri-
butions to the resolution of this problem are also provided by high-quality satellite
photographs, permitting to determine the run-up area. But the velocities of water
flows usually have to be estimated from indirect data.
∼
